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entropy Article

An Operation Reduction Using Fast Computation of an Iteration-Based Simulation Method with Microsimulation-Semi-Symbolic Analysis † Vladimir Mladenovic 1 and Matjaz Debevc 4, * 1 2 3 4

* †

ID

, Danijela Milosevic 1 , Miroslav Lutovac 2 , Yigang Cen 3

Faculty of Technical Sciences Cacak, University of Kragujevac, 32000 Cacak, Serbia; [email protected] (V.M.); [email protected] (D.M.) The School of Electrical Engineering and Computer Science of Applied Studies Belgrade, 11000 Belgrade, Serbia; [email protected] School of Computer and Information Technology, Beijing Jiaotong University, 100044 Beijing, China; [email protected] The Faculty of Electrical Engineering and Computer Science, University of Maribor, 2000 Maribor, Slovenia Correspondence: [email protected] or [email protected]; Tel.: +386-31-346-300 This paper is an extended version of our paper published in Studies in Informatics and Control 2016, 25, 303–312, complemented by new results.

Received: 24 October 2017; Accepted: 11 January 2018; Published: 18 January 2018

Abstract: This paper presents a method for shortening the computation time and reducing the number of math operations required in complex calculations for the analysis, simulation, and design of processes and systems. The method is suitable for education and engineering applications. The efficacy of the method is illustrated with a case study of a complex wireless communication system. The computer algebra system (CAS) was applied to formulate hypotheses and define the joint probability density function of a certain modulation technique. This innovative method was used to prepare microsimulation-semi-symbolic analyses to fully specify the wireless system. The development of an iteration-based simulation method that provides closed form solutions is presented. Previously, expressions were solved using time-consuming numerical methods. Students can apply this method for performance analysis and to understand data transfer processes. Engineers and researchers may use the method to gain insight into the impact of the parameters necessary to properly transmit and detect information, unlike traditional numerical methods. This research contributes to this field by improving the ability to obtain closed form solutions of the probability density function, outage probability, and considerably improves time efficiency with shortened computation time and reducing the number of calculation operations. Keywords: microsimulation-semi-symbolic analysis; iteration-based simulation method; Kummer’s transformation

1. Introduction In general, theoretical, experimental, and computational approaches are the basis for the study of observed phenomena. Every scientific and experimental result is expected to be placed into a function for its use, so the commercial use of products and services, and many engineering uses, emanate from a scientific approach that has been translated into an engineering approach. Emerging developments are posing challenges in information technology [1–3] that include searching large databases [1,2], solving complex processes described by mathematical models, analyzing phenomena in communications in the information space, such as the transmission of wireless signals in urban environments [4–6], and the continuous high speed delivery of information without Entropy 2018, 20, 62; doi:10.3390/e20010062

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stagnation in software engineering [7]. The common feature among all these issues is how to directly obtain results for further processing or exploitation. To address this challenge, complex mathematical tools are used to perform analyses and simulations of the performance of the observed processes and systems. Most often, a classical mathematical analysis does not provide final answers in closed form for complex phenomena, so special functions are used to obtain a solution. When results cannot be obtained with mathematical models, we use numerical methods. Most numerical tools include complex calculations, such as differential and integral equations and algebraic structures, using numerical mathematics algorithms such as Newton-Cotes, Romberg integration, Gauss-Christoffel, trapezium rule, Gauss formulas, etc. [8,9]. Due to this complexity, students may not understand the complete process or system, or cannot perform the method performance analysis to the end within the bounds of the classroom. Engineers and researchers may not have appropriate insight into the impact of the parameters necessary for the effective investigation or design. Additionally, the numerical computation generates a large amount of data, which may sometimes lead to erroneous results [10,11]. These incorrect results may be due to the finite word length in the records, or errors occurring during the shortening of numbers in fractions, for example. These methods do not provide the ability to manipulate with analytic expressions. These issues have been overcome by introducing a new method that treats variables and parameters as symbols. The method is called the iteration-based simulation method (IBSM) [11,12]. In addition to the IBSM that provides symbolic analysis, we allow the analysis to be partially observed through the concept of microsimulation analysis, which additionally enhances the ability to influence the parameters and variables. Also, we created an effective method with fast computation time that, together with the operation reduction, provides accurate results quickly. Computer algebra systems are important tools for these analyses, developments, and research, as they provide a completely new approach for understanding and solving complex cases. In this paper, we present the above methods directly applied to two examples. Both examples require complex analysis and use symbolic closed form expressions, and their numerical analysis is time-consuming. The paper is organized as follows: in Section 2, the problem statement is described. Section 3 illustrates the complete methodology and procedure for applying the method, as well as examples with results in details. In Section 4, the collected results are discussed in detail. 2. Problem Statement A large number of simulations do not guarantee that tolerances will not be exceeded. This is one of the numerous drawbacks of numerical-based tools. Our study had three goals. The first was to solve any analysis in closed form to allow further simplification and manipulation by using an iteration-based simulation method. The second goal was to develop an algorithm to quickly compute the method. Finally, we wanted to reduce the number of operations in the algorithm prior to its implementation. All phases of development and testing were observed by microsimulation-semi-symbolic analysis. The IBSM was developed using the computer algebra system (CAS) to simplify complex algebraic expressions that offers an acceptable reduced analytic form for further manipulation or simulation as a closed form solution as previously published [12]. As integrals are present in the majority of the analyses, we approached the analyses with elementary calculating when the integrals are presented using Riemann sum. The method converts low-complexity implementation into a high-complexity structure. This approach allows implementation in the hardware environment. The CAS performs symbolic mathematical operations and is used in the fields of mathematics and computer science. The CAS is based on algebraic calculations and manipulations performed using the same process as manual derivations. The CAS exclusively includes working with symbols, and numerical calculation is a special case for a CAS. Since symbols are used as variables, CAS deals with symbolic processing. Symbolic processing (SP) involves the development, implementation, and application of algorithms that manipulate and analyze mathematical expressions. CAS provides a deeper understanding and helps students to learn and engineers to simulate and design. The Wolfram language (WL) is the programming language suitable for CAS. WL has the ability to manipulate

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symbolic expressions using a method similar to traditional manual derivation [13]. The WL is characterized by high-performance computing and the generation of compact and short program codes. The goal of IBSM is to introduce a new parameter to obtain a closed form expression. Since the Entropy 2017, 19, 62 3 of 19 iteration is a new parameter, we used a transformation to change the integral into a sum, i.e., a series. For this purpose, usedisRiemann sum transformation for the features the improper integrals. The goalwe of IBSM to introduce a new parameter to obtain a closed formofexpression. Since the is a newwe parameter, weclosed used a transformation to change integral into a sum, i.e., a series. Using iteration this method, obtained form expressions thatthe can be manipulated and simplified, thiscomputation purpose, we used Riemann sum transformation for the features of thewere improper integrals. with a For short time, while reducing the operations. The results tested and verified. Using this method, we obtained closed form expressions that can be manipulated and simplified, The general form of the Riemann integral transformation into a series is given as follows [14,15]: with a short computation time, while reducing the operations. The results were tested and verified. The general form of the Riemann bintegral transformation into a series is given as follows [14,15]: Z n

f (b x )dx =

a



lim

n

∑ f (xi )∆xi



f ( x)dx  k∆x limk→0i=f1( xi )xi

a

x 0

(1)

(1)

i 1

By observing thethe integrals theprevious previous session, we defined types of Riemann By observing integrals in in the session, we defined two typestwo of Riemann sums: a single sums: a single sum and a double sum. We first solved the single sum, then solved the double integrals. sum and a double sum. We first solved the single sum, then solved the double integrals. So, given So, given Equation (1), the Wolfram codeinisFigure shown in Figure 1, value where is iteration the value Equation (1), the Wolfram languagelanguage code is shown 1, where q is the ofqthe in of the thein defined transformation. iteration the defined transformation.

Figure 1. Wolframlanguage language code sum. Figure 1. Wolfram codefor fora aRiemann Riemann sum.

Microsimulation mimics a complex phenomenon by describing its micro-components. Essentially,

Microsimulation mimics a complex phenomenon by describing its micro-components. Essentially, the system is left free to develop without too many constraints and simplifying assumptions [16]. the system is left free to develop without too many constraints and simplifying assumptions [16]. However, when microsimulation is used with only symbolic content, and the particular numerical However, when microsimulation is used symbolic content, and the particular values are changed in the final stage, it is with calledonly microsimulation semi-symbolic analysis (MSSA).numerical We valuesobserved are changed in the of final it calculation is called microsimulation semi-symbolic analysis (MSSA). each element the stage, symbolic through MSSA, which provides faster and better testing and asthe wellsymbolic as a reduction in the operations [17,18]. MSSA directlyfaster calculates We observed eachverification element of calculation through MSSA, whichalso provides and better the verification first run without requiring more simulation testinginand as well as a reduction in theattempts. operations [17,18]. MSSA also directly calculates The next step was the development of an algorithm to allow fast computation. To achieve this, we in the first run without requiring more simulation attempts. treated the expression as a series. As a reminder, a short explanation of the concept of fast computation The next step was the development of an algorithm to allow fast computation. To achieve this, we follows. A series is said to converge slowly if a large number of members of the series need to be treatedadded the expression asaasum series. reminder, a short explanation of the concept of fast computation to determine withAs thearequired accuracy. During the addition of series members using the follows. A series is said to converge slowly if a large number of members of the series need to be member-by-member technique, the process automatically occurs and is interrupted when a selected added criterion to determine a sum with the accuracy. During summation, the additionthe ofabsolute series members using the for error evaluation is required fulfilled. Given the ultimate value of the relationship between the last member and the calculated sum is most often used. This criterion is not member-by-member technique, the process automatically occurs and is interrupted when a selected always especially for additionGiven of a trigonometric series. An error caused an interrupted criterion for reliable, error evaluation is the fulfilled. the ultimate summation, the by absolute value of the summing is always higher than estimated. Conversely, contemporary computing machines can quickly relationship between the last member and the calculated sum is most often used. This criterion add a large number of members in the series. However, due to the limitation on the format of the records is not always reliable, especially for the addition of a trigonometric series. An error caused by an in the registers, a certain number of decimal places are eliminated, which leads to the accumulation of interrupted summing is always higher thaninestimated. computing errors and to completely absurd results the processConversely, of summing.contemporary Therefore, procedures exist machines for can quickly add the a large number members the series. due to the limitation accelerating convergence of aof series, such as in Kummer, Aitken,However, Cesar, and Euler. This paper presentson the method for accelerating convergence of of a series basedplaces on Kummer’s transformation. formatan ofeffective the records in the registers, athe certain number decimal are eliminated, which leads n to the accumulation and to completely absurd the processthen of summing. We adheredoftoerrors two theorems. The first states that ifresults a k in convergences, lim a k  0 . Therefore, The  k  k 1 procedures exist for accelerating the convergence of a series, such as Kummer, Aitken, Cesar, and n n a Euler. second This paper effective method accelerating based on statespresents that if an a and bk areforpositive series, the andconvergence if lim k   ,of (bk aseries 0) , then k   k  b k 1 k 1 k Kummer’s transformation. convergence or divergence occur simultaneously. Kummer’sn transformation, better known as We adhered to two theorems. The first states that if ∑ ak convergences, then lim ak = 0. Kummer’s acceleration method, accelerates the convergence ofk=many series. The method subtracts k→∞ 1 n

n

The second states that if ∑ ak and ∑ bk are positive series, and if lim k =1

k =1

ak

k → ∞ bk

= ρ, (bk 6= 0), then

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convergence or divergence occur simultaneously. Kummer’s transformation, better known as Kummer’s acceleration method, accelerates the convergence of many series. The method subtracts from aagiven givenconvergent convergentseries series a , aand , and another equivalent series bk b k , whose from sum C sum = ∑ Cbkis well ∑ k k another equivalent series ∑ bk , whose







k ≥0

known and finite. transformation is described as: is well known andKummer’s finite. Kummer’s transformation is described as:     ∞ ∞  ∞   ∞ bbk k    bk b k     C + ∑1 1− ρ a k ak  11− ρ aakk =  ρC ∑ ak =akρ ∑ bk +bk ∑ aak k  a k a k kk= 0  k = 0 k 0 k =0 k 0 k =0k 0  0









k 0

(2) (2)

The convergence convergenceof ofthe theright righthand handside sideofofEquation Equation(2)(2)isisfaster faster because 1 −ρρ·b k/akk tends to 0 0 as as kk The because 1− ·bk /a tends to to infinity infinity [19]. [19]. The The complete complete procedure procedureisisshown shownin inFigure Figure2.2. tends

Figure 2. Steps of the speeding up and operation reduction process. Figure 2. Steps of the speeding up and operation reduction process.

The reduction in operations was performed by counting all math operations and functions contained The reduction in operations was performed by performance counting all of math operations functions in the final expressions. Wolfram language allows the direct counting.and Mathematical contained in the final expressions. Wolfram language allows the performance of direct counting. operations and functions in WL can be viewed both symbolically and as commands. Operations are Mathematical operations and functions in WL can be viewed both symbolically and as commands. recognized using the FullForm command, and the counting is performed using the StringPosition Operations are recognized using the FullForm command, and the counting is performed using the command. Since we had sums where the numbers are repeated q times, the WL code for completely StringPosition command. Since we had sums where the numbers are repeated q times, the WL code counting the operations is: for completely counting the operations is: InnerOperations=q*FullForm[aK[z,q];] InnerOperations=q*FullForm[aK[z,q];]

StringPosition[InnerOperations,{"Times","Power","Plus","Rational", "BesselI", "Log","Exp"}]; "BesselI","Log","Exp"}]; The ordersofof Times, Plus, BesselI, Log, and Exp are functions used in close form The orders Times, Plus, BesselI, Log, and Exp are functions used in close form expressions. expressions. Similarly, substituting s for ak[z,q], we obtained the number of operations in the Similarly, substituting s for ak[z,q], we obtained the number of operations in the accelerated algorithm. accelerated algorithm. StringPosition[InnerOperations, {"Times","Power","Plus","Rational",

3. Applications of the Accelerating Procedure and Operation Reduction with Microsimulation 3. Applications of the Accelerating Procedure and Operation Reduction with Microsimulation Semi-Symbolic Analysis Analysis Semi-Symbolic In this this section, section, the the operation operation reduction reduction using using fast fastcomputation computation of ofan aniteration-based iteration-based simulation simulation In method with microsimulation-semi-symbolic analysis was applied to two processing problems to method with microsimulation-semi-symbolic analysis was applied to two processing problems to illustrate the shorter computation time of the algorithm, and to demonstrate the variety of applications illustrate the shorter computation time of the algorithm, and to demonstrate the variety of applications for which which the case with complex calculation is illustrated in the with for the operation operationmay maybe beused. used.AA case with complex calculation is illustrated inexample the example non-coherent Amplitude-Shift Keying (ASK) with shadowing, interference, and correlated noise. The with non-coherent Amplitude-Shift Keying (ASK) with shadowing, interference, and correlated noise. second example treatstreats second-order statistics in the in SCthe macrodiversity system system operating over Gamma The second example second-order statistics SC macrodiversity operating over shadowed Nakagami-m fading channels [20]. Gamma shadowed Nakagami-m fading channels [20]. 3.1. Non-Coherent Amplitude Shift Keying (ASK) with Shadowing, Interference, and Correlated Noise Non-coherent ASK is a modulation scheme used to send digital information between digital equipment and it is shown on Figure 3. Similar part of the system, where real-time estimation is

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3.1. Non-Coherent Amplitude Shift Keying (ASK) with Shadowing, Interference, and Correlated Noise Non-coherent ASK is a modulation scheme used to send digital information between digital Entropy 2017, 19, 62 5 of 19 equipment and it is shown on Figure 3. Similar part of the system, where real-time estimation is needed, can can be be found found in in [21]. [21]. The system without without aa carrier carrier in in needed, The data data is is transmitted transmitted by by the the non-coherent non-coherent system a binary manner. a binary manner.

(ASK) system system with with interference interference ii11(t). Figure 3. Non-coherent amplitude shift keying (ASK)

Shadowing with with interference interference is is one one of of the models used used in in wireless communications Shadowing the most most common common models wireless communications to describe the phenomenon of multiple scattering [21–24]. The basic components of the system system are are to describe the phenomenon of multiple scattering [21–24]. The basic components of the shown in Figure 1. Both shadowing and interference cause strong fluctuations in the amplitude of shown in Figure 1. Both shadowing and interference cause strong fluctuations in the amplitude of the the useful signal. occurs in urban areasis and is described as a log-normal distribution. In our useful signal. This This occurs in urban areas and described as a log-normal distribution. In our analysis, analysis, we performed an outage probability. Transmitting signals using two symbols were observed we performed an outage probability. Transmitting signals using two symbols were observed in the in the non-coherent ASK system [25,26]. The noise, as a narrow-band stochastic process, is correlated non-coherent ASK system [25,26]. The noise, as a narrow-band stochastic process, is correlated and and the coefficient of correlation is denoted by R (R ≠ 1). Mathematically, the noise can be described the coefficient of correlation is denoted by R (R 6= 1). Mathematically, the noise can be described as as(t) ni(t) xi·(t)·cos(ωt) yi(t)·sin(ωt). Thereceiver receiver sheltered, and optical visibility exists toward n = x=i (t) cos(ωt) − y−i (t) ·sin(ωt). The is is sheltered, and no no optical visibility exists toward the i the transmitter, but interference i 1(t) = A1·cos(ωt) is present. If the system sends logical zero, then the transmitter, but interference i1 (t) = A1 ·cos(ωt) is present. If the system sends logical zero, then the signal s0(t) = a0·cos(ωt) has been sent, but if the system sends a logical unit, then the signal s1(t) = ssignal 0 (t) = a0 ·cos(ωt) has been sent, but if the system sends a logical unit, then the signal s1 (t) = a1 ·cos(ωt) ahas 1·cos(ωt) has been sent. The parameters a0 and a1 are the signal elements from which the code words been sent. The parameters a0 and a1 are the signal elements from which the code words are formed. are formed. receiver detects information signaland b0·cos(ωt) and bwith 1·cos(ωt) with envelops z0 and z1 after The receiverThe detects information signal b0 ·cos(ωt) b1 ·cos(ωt) envelops z0 and z1 after passing passing through a transmitting channel. The b m (m = 0, 1) are the elements of the detected signals. The through a transmitting channel. The bm (m = 0, 1) are the elements of the detected signals. The receiver receiverincludes system includes a filter and envelope. detector envelope. In the input, receiver system a filter and detector In the receiver theinput, signalthe is: signal is: rm(t) = bm·cos(ωt) + A1·cos(ωt) + xm·cos(ωt) − ym·sin(ωt) = zm·cos(ωt + φm), m = 0, 1 (3) rm (t) = bm ·cos(ωt) + A1 ·cos(ωt) + xm ·cos(ωt) − ym ·sin(ωt) = zm ·cos(ωt + φm ), m = 0, 1 (3) with envelopes z0 and z1, and phases φ0 and φ1, respectively. generalz0form density function is: with The envelopes and zof andcondition phases φ0joint andprobability φ1 , respectively. 1 , the The general form of the condition joint probability density function is:  x02  x12  y 02  y12  2 R ( x0 x1  y 0 y1 )  1 p( x0 , x1 , y 0 , y1 )  exp (4)  ) ( 2 2 ) ( x x + y y ) 4 21 4 1  R 2  x2 + x2 + y22+ (y12−R 2R 0 1 0 1 0 0 1 1 √ exp − (4) p ( x0 , x1 , y0 , y1 ) = 2) 2 σ 4 1 − R2 2σ2 (1 the − Rset where R is the coefficient of 4π correlation and σ is variance. To ensure of expressions is solved continuously, using the polar coordinates is necessary, as follows: where R is the coefficient of correlation and σ is variance. To ensure the set of expressions is solved z 0 cos  0  as b0 follows: A1 continuously, using the polar coordinatesx0isnecessary, y 0   z 0 sin  0 (5) x0 x= z0z cos cosϕ01 −  bb1 0−A1A1 1 1 y = −z sin ϕ0 y1 0  z1 sin01 (5) x1 = z1 cos ϕ1 − b1 − A1 The next step was determining the condition density function (JPDF). Substituting y1 =joint −z1probability sin ϕ1 Equation (5) into Equation (4), we obtained: The next step was determining the condition joint probability density function (JPDF). Substituting 0, r1/b0, b1, φ0, φ1, A1) = p(x0, y0, x1, y1)·|J| (6) Equation (5) into Equation (4),p(r we obtained: where |J| is Jacobian. A joint probability density function has a log-normal distribution described as [22]: p(r0 , r1 /b0 , b1 , φ0 , φ1 , A1 ) = p(x0 , y0 , x1 , y1 )·|J| (6) 1  pij (b0 , b1 / A1 )  2 2 (b0  A1 )(b1  A1 ) 1  R 2  10 log(b0  A1 )  ai 2  10 log(b1  A1 )  a j 2   exp  2 2 1  R 2     2 R10 log(b0  A1 )  ai 10 log(b1  A1 )  a j   exp  2 2 1  R 2   

(7)

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where |J| is Jacobian. A joint probability density function has a log-normal distribution described as [22]: 1 √ × pij (b0 , b1 /A1 ) = 2πσ2 (b0 + A1 )(b1 + A1 ) 1− R2  2 (10 log(b0 + A1 )− ai )2 +(10 log(b1 + A1 )− a j ) × exp − (7) 2σ2 (1− R2 ) 

× exp

2R(10 log(b0 + A1 )− ai )(10 log(b1 + A1 )− a j ) 2σ2 (1− R2 )



For i = j = 0, the code word 00 was sent; for i = 1 and j = 0, the code word 01 was sent; for i = 0 and j = 1, the code word 10 was sent; and for i = 1 and j = 1, the code word 11 was sent. So: z0 z√ 1 p(z0 , z1 /b0 , b1 , ϕ0 , ϕ1 , A1 ) = × (2π )2 σ4 1− R2 n o 2 2 )(b1 + A1 )−2Rz0 z1 cos( ϕ0 − ϕ1 ) × exp − z0 +z1 −2R(b0 + A12σ × 2 (1− R2 ) o n ϕ0 +z1 cos ϕ1 ) × exp 2(1+ R)(b0 +b1 +2σ2A2 (11)(−zR0 2cos )

(8)

The last expression can be transformed using a modified Bessel function [27] before derivation of the closed form expression: e x cos α =

∞

∑

In ( x ) cos nα

(9)

1 [cos(α + β + γ) + cos(α − β + γ) + cos(α + β − γ) + cos(α − β − γ)] 4

(10)

n=−∞

and applying trigonometric transformation: cos α cos β cos γ = Equation (8) becomes: p(z0 , z1 /b0 , b1 , ϕ0 , ϕ1 , A1 ) = ∞

= C · eC0 ∑

∞

∑

∞

∑ In (C1 ) Im (C2 ) Ik (C3 )× cos[nϕ0 ] cos[mϕ1 ] cos[k( ϕ0 − ϕ1 )]

(11)

n=−∞ m=−∞ k=−∞

where: C=

z0 z√ 1 (2π )2 σ4 1− R2

C0 = −

z0 2 +z1 2 −2R(b0 + A1 )(b1 + A1 ) 2σ2 (1− R2 )

C1 =

(b0 + A1 )(b1 + A1 ) z0 σ 2 (1− R )

C2 =

(b0 + A1 )(b1 + A1 ) z1 σ 2 (1− R )

C3 =

R · z0 z1 σ 2 (1− R2 )

(12)

Using the Bessel identity In (x) = I-n (x), it follows that: p(z0 , z1 /b0 , b1 , A1 ) = 2π 2 C · eC0 (b0 ,b1 ,A1 ) ×

∞

∑

In [C1 (b0 , b1 , A1 )] · In [C2 (b0 , b1 , A1 )] · In [C3 ]

(13)

n =0

The present interference is described with the Rayleigh distribution over the probability density function (PDF) [23,24] as: ( ) A21 A1 p( A1 ) = 2 exp − 2 ; 0 ≤ A1 ≤ ∞ (14) σ 2σ

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To eliminate the interference, performing averaging is necessary for all values of interference A1 . Z∞

pij (z0 , z1 /b0 , b1 ) =

p(z0 , z1 /b0 , b1 , A1 ) · p( A1 )dA1

(15)

0

The integral in Equation (15) is solved using integral: Zπ Zπ

( cos nϕ0 cos mϕ1 cos k( ϕ0 − ϕ1 )dϕ0 dϕ1 =

−π −π Entropy 2017, 19, 62

π 2 , |n| = |m| = |k| 0, n 6= m 6= k

(16) 7 of 19

The distribution is obtained by averaging ϕ0 and ϕ1 for all values between −π and π.    2 , n  m  k cos n 0 cos m1 cos Zπ kZ(π 0  1 ) d 0 d1   p( z0 , z1 /b0 , b1 ) = p(z0 , z1 /b0 , b1 ,ϕ0,0 ,nϕ1 )mdϕ0kdϕ1



(16) (17)

π The distribution is obtained by averaging−πφ− 0 and φ1 for all values between −π and π.   For all code word combinations, distributions of envelopes are obtained by integrating all values p ( z , z / b , b )  p ( z 0 , z(i1 /=b0 0, , b1 ,1;  0 ,j=1 )d0, 01) d1has been sent, marked (17) 0 1code 0 1 word |ij| between b0 and b1 . So, when the with   Hi Hj in Equation (18), and when the same is detected in the input of the receiver marked with Di Dj , all code of word distributions theFor detection thecombinations, signals is described as: of envelopes are obtained by integrating all values between b0 and b1. So, when the code word |ij| (i = 0, 1; j = 0, 1) has been sent, marked with HiHj in Equation (18), Z∞ and when the same is detected in the input of theZ∞receiver marked with DiDj, the detection of the signals P ( D D /H H ) = p ( z , z ) = p(z0 , z1 /b0 , b1 ) · pij (b0 , b1 )db0 db1 (18) is described as: i j i j ij 0 1



0 0 

P ( D D / H i H j )  pij ( z 0 , z1 )  The outage probability is: i j

The outage probability is:

1

Poutage = 1 − ∑ 1 Poutage  1 

 p( z , z / b , b )  p (b , b )db db 0

1

0

1

0

ij

0

1

(18)

1

∑ P( Hi Hj ) P( Di Dj /Hi Hj )

(19)

1

 P( H H ) P( D D / H H ) i =0 j =0

i

i 0

1

0 0

j

i

j

i

j

(19)

j 0

where P( Hi Hj ) = P( Hi ) · P( Hj ) = 12 · 12 = 41 , i = 0, 1, and j = 0, 1. For the outage probability, 1 1 expression 1 Equation and present in theEquation closed (19) form where P(19) ( H i Hrepresents )  P( H i )  Pclosed ( H j )  form   , i = 0, 1, and j = 0,is1. often For thenot outage probability, j 2 2 4 solution. Closed form expression represents an implicit solution that is contained in a mathematical represents closed form expression and is often not present in the closed form solution. Closed form expression [12]. A closed form solution provides a solved problem in terms of functions and expression represents an implicit solution that is contained in a mathematical expression [12]. A closed mathematical operations from a given and generally-accepted set [28]. In other words, a closed form solution provides a solved problem in terms of functions and mathematical operations from a form solution provides an explicit solution to an observed problem, whereas closed form expression given and generally-accepted set [28]. In other words, a closed form solution provides an explicit solution shows implicitproblem, or insufficient solution. to anan observed whereas closed form expression shows an implicit or insufficient solution. From Equation (7), the joint probability is shown shown in inFigure Figure4.4. From Equation (7), the joint probabilitydensity density function function is

Figure 4. 4.Condition joint probability probabilitydensity density function using Wolfram language for shadowing Figure Condition joint function using Wolfram language for shadowing and andinterference. interference.

Figure 5 showsthe themanipulating manipulatingof of Equation Equation (8) (8) and and substituting forfor thethe Figure 5 shows substitutinginto intoEquation Equation(12) (12) changing of coefficients for simplification. changing of coefficients for simplification.

Figure 4. Condition joint probability density function using Wolfram language for shadowing and interference. Entropy 2018, 20, 625 shows the manipulating of Equation (8) and substituting into Equation (12) for the 8 of 19 Figure

changing of coefficients for simplification.

Figure 5. Changing of coefficients for simplification.

Figure 5. Changing of coefficients for simplification. Entropy 2017, 19, 62

8 of 19

Entropy 2017, 19, 62 Entropy 2017, 19, 62 Interference A1

8 of 19

8 of 19 6, per and described describedbyby Wolfram language in Figure Interference is A1present is present perEquation Equation(14) (14) and Wolfram language codecode in Figure 6,

Interference A1 is present per Equation (14) and described by Wolfram language code in Figure 6, Interference A1 is present per Equation (14) and described by Wolfram language code in Figure 6,

Figure 6. Rayleigh distribution for interference coded by Wolfram language.

Figure 6. Rayleigh distribution for interference coded by Wolfram language. Figure 6. Rayleigh distribution for interference coded by Wolfram language.

Rayleigh distribution for interference coded by Wolfram language. AveragingFigure all A1 6. values is necessary, according to Equation (15). The general form of the condition Averaging all Aall values is necessary, according to (15). The general form the condition joint Averaging probability function is defined in Equation (14), and is The described inform Figure 7.of condition 1density A1 values is necessary, according to Equation Equation (15). general of the Averaging alldensity A1 values is necessary, according to Equation (15). The general of7.the7.condition joint probability density function is is defined ininEquation (14),and and described in Figure joint probability function defined Equation (14), isisdescribed in form Figure joint probability density function is defined in Equation (14), and is described in Figure 7.

Figure 7. Log-normal distribution for non-coherent ASK in the presence of shadowing and interference. Figure 7. Log-normal distributionfor fornon-coherent non-coherent ASK ASK in and interference. Figure 7. Log-normal distribution in the thepresence presenceofofshadowing shadowing and interference. s is7.marked variance σ, R is the correlation coefficient, and v is of theshadowing order of the The Figure Log-normal distribution for non-coherent ASK in the presence anditerations. interference. finalization of IBSM obtains closed form expressions of the probability density function, and outage s is marked variance σ, R is the correlation coefficient, and v is the order of the iterations. The sprobability iss marked variance σ, is(Figure the correlation coefficient, v isorder the order ofand theoutage iterations. in ofobtains iterations 8). is marked variance σ, RR is the correlation coefficient, andand v is the of the iterations. The finalization of term IBSM closed form expressions of the probability density function, Thefinalization finalization of IBSM obtains closed form expressions of the probability density function, and outage closed form8). expressions of the probability density function, and outage probabilityofinIBSM term obtains of iterations (Figure probability in term of iterations (Figure 8). probability in term of iterations (Figure 8).

Figure 8. Closed form solution of probability density function (PDFoutage) of a non-coherent ASK system. Figure 8. Closed form solution of probability density function (PDFoutage) of a non-coherent ASK system.

The closed form of PDFoutage in Figure 8 provides the next parameters: iteration q, h0, and h1 are Figure 8. Closed form ofof probability function (PDF ) of a non-coherent system. Figure 8.closed Closed form probability density function outage ) of a non-coherent system. the resolution ofform thesolution iteration, z0in and z1 are envelopes, is(PDF the coefficient of correlation, and is outage The ofsolution PDFoutage Figure 8density provides theRnext parameters: iteration q,ASK h0,ASK and h1σare variance. This expression cannot be manually obtained by using numerical tools. The resultant closed the resolution of the iteration, z0 and z1 are envelopes, R is the coefficient of correlation, and σ is The closed form outage in 8 ready provides the next parameters: iteration q,the h0viewpoint , and h1 are form solution PoutageofisPDF ancannot expression that is for further processing. variance. Thisof expression beFigure manually obtained by using numericalAccordingly, tools. The resultant closed the resolution of the iteration, z 0 and z 1 are envelopes, R is the coefficient of correlation, and σ is is an insight into the parameters and variables that participate in obtaining all the features of this case form solution of Poutage is an expression that is ready for further processing. Accordingly, the viewpoint variance. This expression cannot be manually obtained by using numerical tools. The resultant closed study. Drawing the characteristics is now possible, but this calculation would take too long, regardless of is an insight into the parameters and variables that participate in obtaining all the features of this case the chosen accuracy. On the other hand, for greater accuracy, a number of iterations is required, form solution of P outage is an expression that is ready for further processing. Accordingly, the viewpoint study. Drawing the characteristics is now possible, but this calculation would take too long, regardless of is not beneficial for the thisother form of expression. is which an insight into the parameters and variables participate in obtaining features this case the chosen accuracy. On hand, for that greater accuracy, a number all of the iterations is of required,

Entropy 2018, 20, 62

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The closed form of PDFoutage in Figure 8 provides the next parameters: iteration q, h0 , and h1 are the resolution of the iteration, z0 and z1 are envelopes, R is the coefficient of correlation, and σ is variance. This expression cannot be manually obtained by using numerical tools. The resultant closed form solution of Poutage is an expression that is ready for further processing. Accordingly, the viewpoint is an insight into the parameters and variables that participate in obtaining all the features of this case study. Drawing the characteristics is now possible, but this calculation would take too long, regardless of the chosen accuracy. On the other hand, for greater accuracy, a number of iterations is required, which is not beneficial for this form of expression. Finally, the closed form solution of Poutage is shown in Figure 9. Entropy 2017, 19, 62

9 of 19

FigureFigure 9. Closed outage probability of a non-coherent ASK system with 9. Closedform form solution solution ofof outage probability Poutage ofP aoutage non-coherent ASK system with shadowing and interference. shadowing and interference. In our case, a member ak represents a general member of the series in Poutage, from the closed form

In our case, a member ak represents a general member of the series in Poutage , from the closed form solution in Figure 9. solution inConvergence Figure 9. testing of the ak verified that: Convergence testing of the ak verified that: lim ak  0 k q q

(20)

lim ak = 0 Convergence testing was performed with that 0 ≤ R < 1, σ > 0, z ≥ 0, and q ≥ 1. k →assumptions q The selection of the auxiliary function is one of the most important aspects of the MSSA [29]. In q→∞

(20)

testing many series, the authors of this paper highlighted the series that shows the best performance to accelerate convergence, meaning a shorter computation time with the optimum number of iteration. Convergence testing was performed with assumptions that 0 ≤ R < 1, σ > 0, z ≥ 0, and q ≥ 1. Comparative analysis of different auxiliary series can be the subject of particular surveys, and the reader(s) are encouraged to do so. Therefore, in our case, the auxiliary series is: 

C

1

k 2 5

k 1

(21)

k 1

The series converges to 2log2. To fully use Equation (2), we made a minor modification to the member bk, with respect to the convergence theorems that have been mentioned above. The new

Entropy 2018, 20, 62

10 of 19

The selection of the auxiliary function is one of the most important aspects of the MSSA [29]. In testing many series, the authors of this paper highlighted the series that shows the best performance to accelerate convergence, meaning a shorter computation time with the optimum number of iteration. Comparative analysis of different auxiliary series can be the subject of particular surveys, and the reader(s) are encouraged to do so. Therefore, in our case, the auxiliary series is: ∞

C=

∑

k =1

1

(21)

k 5 2k −1

The series converges to 2log2. To fully use Equation (2), we made a minor modification to the member bk , with respect to the convergence theorems that have been mentioned above. The new member becomes bk → ak + ck , so: ∞ Entropy 2017, 19, 62

∑

s =19, 62 Entropy 2017,

k =0

∞

ak = ρ ∑ ak + k =0

∞

∑

k =0

 1−ρ

  ∞  ak + ck a + ck ak = ρC + ∑ 1 − ρ k ak ak ak k =0

10 of 19 10 of 19

(22)

      a  ck  a  ck  a k  C   1   k a s   a k    a k   1   k (22)   member      c  k Poutage marked as ak a c a a where ck is general term (21). We obtain the general k k 0 Equation  kin k 0 k 0  k 0 k 1   a k  C  1   k a k k aof s ak   ak  (22) k   a k the next step a k  we derived the term ρ k 0 k 0 Following k 0  in MSSA, in Figure 10, separating itk 0from Figure 9.  where ck is general term in Equation (21). We obtain the general member of Poutage marked as ak in Figure (Figurewhere ck is general term in Equation (21). the Wenext obtain theingeneral of Pthe outage marked as ak in 10,11). separating it from Figure 9. Following step MSSA, member we derived term ρ (Figure 11).Figure 10, separating it from Figure 9. Following the next step in MSSA, we derived the term ρ (Figure 11).

 

 

 

 

Figure 10. General term in a series of Poutage marked as ak. Figure 10. General term in a series of Poutage marked as ak.

Figure 10. General term in a series of Poutage marked as ak .

Figure 11. The element Kummer’s transformation ρ. Figure 11. The element Kummer’s transformation ρ.

Figure 11. The element Kummer’s transformation ρ. We checked that the value ρ tends to 1 after convergence testing. The quicker computation was We checked that the how valuemuch ρ tends to 1 after testing. The performed by assuming iteration is convergence required to calculate the quicker outage computation probability Pwas outage performed how much iteration is required to calculate the outage probability Poutage obtained byby theassuming IBSM. Otherwise, a large number of iterations are required to calculate the closest obtained by of thePoutage IBSM. a largeisnumber of iterations arethe required to calculate the closest exact values , butOtherwise, the computation time consuming. Then, resulting Poutage equalizes with Poutage, but is time consuming. Then, thepoint resulting Poutage for equalizes with aexact newvalues series of obtained by the computation Kummer's transformation, and performs matching the various avalues new series by followed the Kummer's transformation, and performs pointAfter matching for verification the various of theobtained envelopes, by a new reduced number of iterations. that, the values of the envelopes, followed by a new reduced number of iterations. After that, the verification

Entropy 2018, 20, 62

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We checked that the value ρ tends to 1 after convergence testing. The quicker computation was performed by assuming how much iteration is required to calculate the outage probability Poutage obtained by the IBSM. Otherwise, a large number of iterations are required to calculate the closest exact values of Poutage , but the computation is time consuming. Then, the resulting Poutage equalizes with a new series obtained by the Kummer's transformation, and performs point matching for the various values of the envelopes, followed by a new reduced number of iterations. After that, the verification of the obtained results was performed by checking the relative error, which determined the degree of adjustability of the algorithm [29]. Finally, we checked the number of operations of calculations in the expression in Figure 9, and then obtained a reduced number of operations with a new decreased number of iterations. After all symbolic derivations, we used closed form solutions to directly obtain results in the first attempt. To obtain concrete numerical results, we needed to set the initial parameters. We supposed that the closest exact value was obtained after 500 iterations by using the outage probability Poutage in Figure 9, and the resolution of the iteration was h0 = 0 and h1 = 1. We also used z0 = z1 = z to simplify the analysis. The next step was calculating the new numbers of iterations that are reduced for various values of the envelope z. This was performed using the command FindRoot[s==Poutage,{q,1}]. s is a new expression obtained by Kummer’s transformation in Equation (22), and Poutage is a closed form solution in Figure 8. We took the range of values z = {1, 15} for a concrete case [29]. Experiments were performed for various values of the coefficient of correlation R (R = 7/10,8/10) and the variance σ (σ = 2, 3). All calculations were performed with a precision of 10−6 . All tests were performed on a PC with: Intel® Core™ i5-6500 CPU@ 3.2 GHz, 8 GB RAM, 64-bit Operating System, Windows 10, and Mathematica Wolfram 11.1. The reduced number of iterations are shown in Table 1. Table 1. Reduced number of iterations. z

q1 R = 7/10; σ = 2

q2 R = 7/10; σ = 3

q3 R = 8/10; σ = 2

q4 R= 8/10; σ = 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

25 20 19 18 17 17 18 18 20 21 23 27 29 32 36

32 27 25 24 23 23 23 23 24 25 26 27 28 30 31

21 16 14 12 11 10 9 9 9 9 10 11 12 14 16

30 25 22 21 20 20 19 19 20 19 20 21 21 22 23

In Figure 12, the changing iteration values in terms of the envelope z are shown for the accelerated algorithm. Notably, the reduced number of iterations is not the same for each envelope value. The minimum number of iterations is z = 10, where the value provides a true detection. However, the number of iterations is in range of 9 to 35 if we observe the total range of the envelope, which is a significant reduction compared to the original 500 iterations.

In Figure 12, the changing iteration values in terms of the envelope z are shown for the accelerated algorithm. Notably, the reduced number of iterations is not the same for each envelope value. The minimum number of iterations is z = 10, where the value provides a true detection. However, the number of iterations is in range of 9 to 35 if we observe the total range of the envelope, which is a Entropy 2018, 20, 62 12 of 19 significant reduction compared to the original 500 iterations.

Figure envelope z. z. Figure 12. 12. The The number number of of iterations iterations in in term term of of envelope

Since the absolute error is not precisely characterized by accuracy, the relative error is used as: 12 of 19 error is not precisely characterized by accuracy, the relative error is used 12 of 19as:

Entropy 2017, 19, 62 Since 2017, the absolute Entropy 19, 62

s − Poutage Poutage δ= ss  P outage   Poutage Poutage P outage

(23) (23)

(23)

Relative errors do do notnot exceed more value asititisisshown showninin Figure This indicates Relative errors exceed morethan than10% 10%of ofthe the value value as as Figure 13.13. This indicates Relative errors do not exceed more than 10% of the it is shown in Figure 13. This indicates that that the algorithm is quite accurate. In Figure 14, the comparative characteristics of P and outage the algorithm algorithm is is quite quite accurate. accurate. In In Figure Figure 14, 14, the the comparative comparative characteristics characteristics of of P Poutage outage and ss are ares are that the and shown. TheThe accelerated algorithm e,approx. . shown. The accelerated algorithms sis s is ismarked markedas as P Pe,approx e,approx. shown. accelerated algorithm marked as P

Figure 13. Relativeerror errorfunctions functions in in term of the envelope z. z. Figure 13.13. Relative in term termof ofthe theenvelope envelope Figure Relative error functions z.

Figure 14. Comparative characteristics of Poutage and accelerated outage probability s.

Figure Comparative characteristicsof ofPPoutage and accelerated outage probability s. Figure 14. 14. Comparative characteristics outage and accelerated outage probability s.

The total total calculation calculation of of formula formula P Poutage outage required 1193.97 s, or 19 min and 54 s, so the average time The required 1193.97 s, or 19 min and 54 s, so the average time per iteration was 70.2335 s. The sped up algorithm’s total total calculation calculation time time for for the the accelerated accelerated formula formula per iteration was 70.2335 s. The sped up algorithm’s was 1.25 s, so the average time per iteration was 0.0735294 s. Wolfram language code for time consumed was 1.25 s, so the average time per iteration was 0.0735294 s. Wolfram language code for time consumed is: Table[Timing[N[Poutage]],{z,15}] Table[Timing[N[Poutage]],{z,15}] // // Total. Total. Command Command Table Table provides provides aa calculation calculation is: for any value of envelope z, and command Timing provides the exact time of calculation. Command for any value of envelope z, and command Timing provides the exact time of calculation. Command Total summarizes total time per envelope. Similarly, changing the parameter Poutage with s s for for Total summarizes total time per envelope. Similarly, changing the parameter Poutage with

Entropy 2018, 20, 62

13 of 19

The total calculation of formula Poutage required 1193.97 s, or 19 min and 54 s, so the average time per iteration was 70.2335 s. The sped up algorithm’s total calculation time for the accelerated formula was 1.25 s, so the average time per iteration was 0.0735294 s. Wolfram language code for time consumed is: Table[Timing[N[Poutage]],{z,15}] // Total. Command Table provides a calculation for any value of envelope z, and command Timing provides the exact time of calculation. Command Total summarizes total time per envelope. Similarly, changing the parameter Poutage with s for the accelerated algorithm in the previous WL command line provides the time consumed for fast computation. Our algorithm is accelerated as: Ratio =

time( Poutage ) 1193.97 = ≈ 955times time(s) 1.25

(24)

Figure 15 shows the number of operations in terms of the number of iterations q for fast computation. The number of iterations is fixed at q = 500 for Poutage because we initially assumed that this number Entropy 2017, 19,of62iterations was satisfied for the closest exact value of Poutage . The number of operations 13 of 19 for fast computation of IBSM is less than Poutage . For 500 iterations, we counted 120,000 math operations Poutage . The. The number of math operations changes in the range of 9000 to 34,000, which is the result of for Poutage number of math operations changes in the range of 9000 to 34,000, which is the result variety in in thethe number of of iterations forfor fast computation. of variety number iterations fast computation.

Figure 15. 15. Number Number of of operations operations in in terms terms of of number number of of iterations iterations qq for for fast fast computation. computation. The The number number Figure outage . of iterations is fixed with q = 500 for P of iterations is fixed with q = 500 for Poutage .

3.2. Second-Order Second-Order Statistics Statistics in in Wireless Wireless Channels Channels 3.2. The level level crossing crossing rate rate (LCR) (LCR) and and the the average average duration duration of of fade fade (ADF) (ADF) are are important important second-order second-order The statistical characteristics describing the fading channel in mobile communications. These values are are statistical characteristics describing the fading channel in mobile communications. These values suitable for forfor analyzing their performance. In suitable for designing designing mobile mobileradio radiocommunication communicationsystems systemsand and analyzing their performance. digital telecommunications, a sudden drop in the value of the received signal directly leads to a drastic In digital telecommunications, a sudden drop in the value of the received signal directly leads to the probability of error. For optimizing the codingthe system required correct errors, the aincrease drastic in increase in the probability of error. For optimizing coding systemtorequired to correct number of times the received signal passes through the given level in time and how long, on average, errors, the number of times the received signal passes through the given level in time and how the signal is belowthe thesignal specified level must be known. Themust LCR be andknown. ADF are theLCR appropriate measures long, on average, is below the specified level The and ADF are the closely related to the quality of the received signal [24]. appropriate measures closely related to the quality of the received signal [24]. The LCR LCR of of signal signal Z(t), Z(t), marked marked as as N NZ(z), is defined as the signal speed crossing through level z The Z (z), is defined as the signal speed crossing through level z with a positive derivative at the intersection point z. The The ADF, ADF, marked markedas asTTZ(z), (z), represents represents the the mean mean with a positive derivative at the intersection point z. Z time for for which which the the signal signal overlay overlay is is below below the the specified specified zz level. level. time The LCR at envelope z is mathematically defined by [22]: 









N Z ( z )  z p  ( z , z )d z 0

(25)

zZ





where z is the envelope of the received signal, z is its derivative in time, and p  ( z, z ) is the joined zZ

probability density function. The average fade duration (AFD) is determined as [22]:

TZ ( z) 

FZ ( z  Z )

(26)

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The LCR at envelope z is mathematically defined by [22]: NZ (z) =

Z∞

•

•

•

zp • (z, z)dz

(25)

zZ

0

•

•

z)62is the joined where z is the envelope of the received signal, z is its derivative in time,Entropy and p2017, • ( z,19, zZ

probability density function. The average fade duration (AFD) is determined as [22]: 22.

Borhani, A.; Patzold, M. Modeling of Vehic In Proceedings of the 2012 IEEE Vehicular T F (z ≤ Z ) TZ (z) = Z 3–6 September 2012; (26) pp. 1–5, doi:10.1109/VT NZ (z) 23. Patzold, M. Mobile Radio Channels, 2nd ed.; J where FZ (z ≤ Z) represents the probability that the signal level Z(t) is less than 51747-5. the level z. Evaluation and calculation of LCR and ADF are trivial in an environment where24. no large reflections existsChannels; John Wiley Patzold, M. Mobile Fading with a large number of transmission channels and shadowing, which simplifies the mathematical 25. Lutovac, M.; Mladenovic, V.; Lutovac, M. D Traffic Control Using OFDM and Computer description of the distribution of the signal. However, in complex environments, obtaining LCR 26. Martinez, Martinez, and ADF characteristics is time-consuming. An example of a complex environment is W.; described in A. Computational London, UK, 2015; ISBN Stefanovic et al. [20]. In this example, the LCR and ADF expressions were obtained. Their analytical 9781466592735. 27. Thus, Andrews, Askey, R.;isRoy, R. Special Functi shapes are closed forms, but the complexity shows a long computation time. the G.; LCR value University Press: Cambridge, UK, 2001; pp. 2 normalized by the Doppler shift frequency fd [20] through Equation (15): 28. Borwein, J.M.; Crandall, R.E. Closed forms:   q 55, doi:10.1090/noti936. M1 −1 NZ (z) N1 m1 2π z M1 −1 = × 29. Mladenovic, V.; Makov, S.; Cen, Y.G; Luto r1 m1 fd Γ( M1 )Γ(c1 )Γ(c2 )  q  M1 −c1 −c2 +k−1/2 ∞ Study of Non-coherent ASK 2 N1 m1 zMethod—Case (Ω01 +Ω02 ) ( N1 m1 z/r1 ((1/Ω01 )+(1/Ω02 ))) ×∑ K( M1 +c1 +c2 +k−1/2) 2 30. r Ω + http://reference.wolfram.com c1 k + c2 Ω Available online: 02 1 01 c2 (1+ c2 ) Ω Ω k =0

k

01

02

  M2 q M −1 2π + Γ( M z)Γ(2c )Γ(c ) N2r2m2 m2 × 2 2 1 M2 −c1 −c2 +k−1/2 ∞ 2 ( N2 m2 z/r2 ((1/Ω01 )+(1/Ω02 )))

×∑

c

k + c2

c2 (1+c2 )k Ω011 Ω02

k =0

(27)

 q  N2 m2 z(Ω01 +Ω02 ) K( M2 +c1 +c2 +k−1/2) 2 r2 Ω Ω02 01

© 2017 by the authors. S terms and conditions of (http://creativecommons.org

where Γ(x) denotes the Gamma function, Mi is (mi Ni2 )/ri , mi is the Nakagami-m fading severity parameter, Ni denotes the number of identically assumed channels at each microlevel, ri is related to the exponential correlation ρi , ci denotes the order of Gamma distribution, Ώ0i is related to the average powers of the Gamma long-term fading distributions, and Kv (x) is the modified Bessel function of the second order. Similarly, the AFD is obtained as [20] per Equation (16): FZ (z≤ Z ) NZ (z) 2( N m /r ) M1 Pz ( Z ) = Γ( M )1Γ(c1 )Γ1(c ) M × 2 1 1 1 c1 +c2 ++l −k− M1 k ∞ ∞  2 ( N1 m1 z/r1 ((1/Ω01 )+(1/Ω02 ))) N1 m1 ∑ ∑ c1 l + c2 r1 c ( 1 + c ) Ω Ω 2 2 k =0 l =0 k 01 02

TZ (z) =

 q  N1 m1 z(Ω01 +Ω02 ) K(c1 +c2 +l −k− M1 ) 2 + r Ω Ω02 1

01

(28)

) M2

2( N2 m2 /r2 × Γ( M2 )Γ(c1 )Γ(c2 ) M2 ∞

∞

∑ ∑

k =0 l =0



N2 m2 r2

k

( N1 m1 z/r1 ((1/Ω01 )+(1/Ω02 ))) c

c1 +c2 +l −k− M2 2

l + c2

c2 (1+c2 )k Ω011 Ω02

 q  N1 m1 z(Ω01 +Ω02 ) K(c1 +c2 +l −k− M2 ) 2 + r Ω Ω02 1

01

As in the previous example, we defined a general term ak from Equation (27), shown in Figure 16.

2( N 2 m2 / r2 ) 2  ( M 2 )(c1 )(c2 ) M 2 



 k 0

l 0

 N 2 m2   r2

  

k

N m z / r ((1 /  1

1

1

)  (1 /  02 ))  c2 (1  c2 ) k  c011  l02c2 01

c1c2 l k M 2 2

 N1 m1 z ( 01   02 )   K c1c2 l k M 2   2   r1 01 02  

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As in the previous example, we defined a general term ak from Equation (27), shown in Figure 16.

Figure 16. 16. General General term term of of the the level level crossing crossingrate rate(LCR) (LCR)marked markedas asaak.. Figure k

Using the expression in Figure 17, we derived the term ρ that tends to 1 when q → ∞. in Figure 17, we derived the term ρ that tends to 1 when q → ∞.

Using expression Entropy 2017, the 19, 62

15 of 19

Figure17. 17. The The element elementKummer’s Kummer’s transformation transformation ρ. ρ. Figure

In this case, Equations (27) and (28) have already been provided in advance in a closed form where the iteration parameter q is present, so applying the IBSM would be excessive. To compute the closest exact values of LCR and AFD, 100 iterations were required in Stefanovic et al. [20]. Using Kummer’s transformation, both LCR and AFD were calculated in the first iteration. All computations were performed using the values of m = 1, L = 2, Ώ = 1, c = 2, and R = 1/5. An auxiliary series was used:

29. 30.

55, doi:10.1090/noti936. Mladenovic, V.; Makov, S.; Cen, Y.G; Lutovac, M. Fast Computation of the Method—Case Study of Non-coherent ASK with Shadowing. Serbian J. Electr. Available online: http://reference.wolfram.com/language/ref/EllipticTheta.html (ac

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© 2017 by the authors. Submitted for possible open acc terms and conditions of the Creative Commons At In this case, Equations (27) and (28) have already been provided in advance in a closed form where (http://creativecommons.org/licenses/by/4.0/).

the iteration parameter q is present, so applying the IBSM would be excessive. To compute the closest exact values of LCR and AFD, 100 iterations were required in Stefanovic et al. [20]. Using Kummer’s transformation, both LCR and AFD were calculated in the first iteration. All computations were performed using the values of m = 1, L = 2, Ώ = 1, c = 2, and R = 1/5. An auxiliary series was used: ∞

C=

∑ e−k

2

(29)

k =1

The series C converges to (1/2)·(ϑ3 (0, e−1 )–1), where ϑa (u, x), (a = 1, . . . ,4) is the theta function, defined as [30]: ∞

ϑ3 (u, x ) = 1 + 2 ∑ x k cos(2k · u) 2

(30)

k =1

Figure 18 shows the comparative characteristics of LCR and accelerated LCR. The deviation of the accelerated series is small in relation to the original series, and the relative error is shown in Figure 19, 2017, 19, 62 of envelope –35 ≤ z ≤ 30. 16 of 19 in theEntropy specified range

Figure 18. 18. Comparative LCRand andaccelerated accelerated LCR. Figure Comparativecharacteristics characteristics of of LCR LCR.

Figure 19. Relative error function of LCR in term of the envelope z. Figure 19. Relative error function of LCR in term of the envelope z.

The total calculation of the LCR formula required 30.6563 s, so the average time per iteration

The calculation ofcalculation the LCR formula 30.6563 s, sointhe time per iteration was total 0.437946 s. The total time withrequired the sped up algorithm the average accelerated formula was was 0.437946 calculation time with sped up algorithm in is the accelerated 1.53125 s,s.soThe the total average time per iteration wasthe 0.021875 s. Our algorithm accelerated as: formula was 1.53125 s, so the average time per iteration s. Our algorithm is accelerated as: timewas ( LCR0.021875 ) 30.6563 Ratio 

orig

time( LCRaccelerated )



30.6563  20times 1.53125

(31)

time( LCRorig ) 30.6563 Ratio = = ≈ 20times (31) time ( LCR accelerated ) (N1.53125 Z) in terms of the number of iterations q for Figure 18 shows the number of operations of LCR fast computation. The number of iterations was fixed at q = 100 for LCRorig because we initially assumed that this number of iterations satisfied the closest exact value of LCRorig. For 100 iterations, we counted 20,200 math operations for LCRorig. The number of math operations was 1184 for LCRaccelerated calculated in the first iteration using fast computation. Using the same method, the AFD was obtained by applying Equation (22). Figure 20 shows the comparative characteristics of AFD and accelerated AFD. A small deviation in the range of −35 ≤ z ≤ −28 was observed, perceived through the relative error in Figure 21.

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Figure 18 shows the number of operations of LCR (NZ ) in terms of the number of iterations q for fast computation. The number of iterations was fixed at q = 100 for LCRorig because we initially assumed that this number of iterations satisfied the closest exact value of LCRorig . For 100 iterations, we counted 20,200 math operations for LCRorig . The number of math operations was 1184 for LCRaccelerated calculated in the first iteration using fast computation. Using the same method, the AFD was obtained by applying Equation (22). Figure 20 shows the comparative characteristics of AFD and accelerated AFD. A small deviation in the range of −35 ≤ z ≤ −28 was observed, perceived through the relative error in Figure 21. The total calculation of formula AFDorig required 19,553.1 s, or 5 h and 25 min, so, the average time per iteration was 279.33 s, or 4 min and 19.33 s. The sped up algorithm total calculation time with the accelerated formula was 1.29688 s, so, the average time per iteration was 0.0185268 s. An obvious difference in the time calculation exists because the number of sums for AFD increased in Equation (27), where we have sums for k, l, and q. In this case, our algorithm is accelerated as:

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Ratio =

time( AFDorig ) 19, 553.1 = ≈ 15 · 103 times time( AFDaccelerated ) 1.29688

(32) 17 17 of of 19 19

For 100 iterations, iterations, we we counted counted the the344 344×× 10 10666 math operations operations for for AFD AFDorig orig orig.. The The number of math accelerated operations was was 5619 5619 for for LCR LCRaccelerated accelerated calculated operations calculatedininthe thefirst firstiteration iterationfor forfast fastcomputation. computation.

Figure Figure 20. 20. Comparative Comparative characteristics characteristics of of the the average average fade fade duration duration (AFD) (AFD) and and accelerated accelerated AFD. AFD.

Figure Figure 21. 21. Relative Relative error error function function of of AFD AFD in in terms terms of of envelope envelope z. z.

4. Conclusions This paper presents a new method to accelerate the computation and reduce the number of calculation operations in the iteration-based simulation method. The method was developed to simulate the systems and processes when obtaining mathematical formulas in the final closed form is not possible. Often, many phenomena show that closed form expressions and simulations are executed with numerical based tools. In these cases, the users do not have insight into the phenomena

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4. Conclusions This paper presents a new method to accelerate the computation and reduce the number of calculation operations in the iteration-based simulation method. The method was developed to simulate the systems and processes when obtaining mathematical formulas in the final closed form is not possible. Often, many phenomena show that closed form expressions and simulations are executed with numerical based tools. In these cases, the users do not have insight into the phenomena that affect the flow of processes, which can lead to incorrect assumptions and results. The method provides insight into processes and systems using symbolic processing, with significant acceleration and reduction in the number of computation operations required. For symbolic derivation, the computer algebra system was used, and Kummer’s transformation was used to shorten the computation time. The complete method to reduce the number of operations and shorten the computation time was illustrated in two examples. Both cases require complex and time-consuming calculations. Due to the large number of operations, the memory resources can also play a significant role in the speed of the calculation. The acceleration of the algorithm and the reduction of the number of operations significantly affected efficiency in terms of time savings and the rapid production of results. The method can be used in many fields where fast computation in one-step simulation runs is required. Acknowledgments: This paper is partially funded by Serbian Ministry of Education, Science and Technological Development within the project No. TR 32023. Author Contributions: Vladimir Mladenovic developed algorithms and performed testing; Yigang Cen and Vladimir Mladenovic proposed the field of application and microsimulation concept; Miroslav Lutovac wrote Wolfram language code involving by the symbolic processing; Danijela Milosevic and Matjaz Debevc introduced the semi-symbolic concept combining symbolic and numerical data, analyzed data, and performed their verifications; Vladimir Mladenovic and Matjaz Debevc wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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